Evaluate even-dimensional one-loop Witten diagrams in momentum space

Determine the closed momentum-space function class, branch prescription, and symbol alphabet for one-loop Witten diagrams in even boundary dimensions, where the Appell-$F_4$ reduction is algebraic but not rational and the remaining spectral integrals resist elementary evaluation.

Background

At one loop, radial integrations reduce to Appell-F4F_4 functions. In odd boundary dimensions the relevant expression becomes rational, but in even dimensions it contains a half-integer power of a quartic denominator, leaving nontrivial square-root structure in the spectral integrations.

The review notes that position-space calculations suggest polylogarithmic answers, but the direct momentum-space analytic form, its branch structure, and its generalization beyond one loop remain unresolved.

References

In even $d$ the collapse is only algebraic, a half-integer power of the same quartic, and the spectral integrals no longer reduce to rational algebra: the open frontier.

— Cosmological Correlators from Scattering Amplitudes: A Review  (2609.19252 - Albayrak et al., 16 Sep 2026) in Section 5.3, “Appell $F_4$ and the odd-$d$ collapse”